Stock price a chaotic revelation?

For a process x(t), define survivability ratio P(t) = chance of finding x(t) within a predetermined range of x(0); Brownian motion has P(t) = t^(-1/2). For chaotic systems with Cantori trapping mechanisms, P(t) also follows a power law P(t) = t^(-p), p=1+α, with 0<α<1.

Now, assume s(t) is the drift-adjusted part of the stock price while S(t) is the unadjusted (original) stock price. In other words:

d(lnS(t)) = μ dt + ds(t),

we might be able to check the index α for the process s(t), by sampling a lot of (t=0).

Also Re: self-criticality; people have discussed (with mixed opinions) the relationship between financial markets and earthquakes (Gutenberg-Richter law). It might be worthwhile to check the survivability function of fBm. See also 1/f noises.